According to the standard model of particle physics, fundamental constants, at least their low-energy limits, should be independent of time and location. Many fundamental constants of physics and astronomy are well defined locally. However, this ``constancy of constants'' may not necessarily hold over the largest spatial and temporal scales, which are inaccessible by geological or astronomical studies of nearby targets. Now, advances in observational sensitivity make it possible to measure fundamental constants for sources in the distant Universe, viewed at times billions of years ago and preceding the formation of the solar system. I will present a summary of measurements obtained so far, introducing observational methods, but mainly focusing on achieved accuracies, uncertainties, and future prospects of this rapidly expanding field of research.